9 N ov 1 99 9 Non - Invariant Ground States , Thermal Average , and Generalized Fermionic Statistics

We present an approach to generalized fermionic statistics which relates the existence of a generalized statistical behaviour to non-invariant ground states. Considering the thermal average of an operator generalization of the Heisenberg algebra, we get an occupation number which depends on the degree of mixing between symmetric and antisymmetric sectors of the ground state. A natural prescription is given for the construction of a supersymmetric statistics. We also show that the structure of the vacuum, and therefore the statistical behaviour of the system, can be accounted for in terms of a second-order phase transition.